Negative Exponents: Rules, Simplification, and Practice

Exponents & Exponential Functions

Negative Exponents

Negative exponents are a fundamental concept in algebra that allow us to express reciprocals and simplify expressions. Understanding how to rewrite and simplify expressions with negative exponents is essential for manipulating algebraic expressions and solving equations.

• Negative Exponent Rule: For any nonzero number a and integer n, $a^{-n} = \frac{1}{a^n}$.

• Rewriting Negative Exponents: Move the base with a negative exponent to the denominator (or numerator) and change the exponent to positive.

• Zero Exponent Rule: Any nonzero number raised to the zero power is 1: $a^0 = 1$.

Example: Rewrite $x^{-5}$ using positive exponents.

• $x^{-5} = \frac{1}{x^5}$

Example: Simplify $\frac{w^{-7}}{w^9}$.

• Apply the quotient rule: $w^{-7} \div w^9 = w^{-7-9} = w^{-16}$

• Rewrite with positive exponents: $w^{-16} = \frac{1}{w^{16}}$

Rules for Simplifying Exponents

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power of a Power Rule: $(a^m)^n = a^{mn}$

• Power of a Product Rule: $(ab)^n = a^n b^n$

• Power of a Quotient Rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Example: Simplify $(3x)^{-3}$.

• Apply the negative exponent rule: $(3x)^{-3} = \frac{1}{(3x)^3} = \frac{1}{27x^3}$

Practice Problems

Rewrite each expression using positive exponents only:

1. $x^{-7} = \frac{1}{x^7}$

2. $5x^2y^{-3} = 5x^2 \cdot \frac{1}{y^3} = \frac{5x^2}{y^3}$

3. $-4a^{-2}b^{-2} = -4 \cdot \frac{1}{a^2} \cdot \frac{1}{b^2} = -\frac{4}{a^2b^2}$

4. $(ab^2)^{-4} = \frac{1}{a^4b^8}$

5. $-8(x^{-3}y^4)^{-5} = -8 \cdot (x^{-3})^{-5} \cdot (y^4)^{-5} = -8x^{15}y^{-20} = -\frac{8x^{15}}{y^{20}}$

Example: Simplify $\frac{x^{-2}y^4}{x^5y^{-3}}$.

• Apply the quotient rule: $x^{-2-5}y^{4-(-3)} = x^{-7}y^{7}$

• Rewrite with positive exponents: $\frac{y^7}{x^7}$

Challenge Problem

Simplify completely:

• $\left(\frac{x^{-3}y^{-2}}{x^{-4}y^{-4}}\right) \cdot \left(\frac{x^{-1}y^{-7}}{x^{-2}y^{-2}}\right)$

• First, simplify each fraction:

  • $\frac{x^{-3}y^{-2}}{x^{-4}y^{-4}} = x^{-3-(-4)}y^{-2-(-4)} = x^{1}y^{2}$

  • $\frac{x^{-1}y^{-7}}{x^{-2}y^{-2}} = x^{-1-(-2)}y^{-7-(-2)} = x^{1}y^{-5}$

• Multiply: $x^{1}y^{2} \cdot x^{1}y^{-5} = x^{2}y^{-3}$

• Final answer: $\frac{x^2}{y^3}$

Summary Table: Exponent Rules

Rule Name	Formula	Example
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-3} = \frac{1}{x^3}$
Product Rule	$a^m \cdot a^n = a^{m+n}$	$x^2 \cdot x^5 = x^7$
Quotient Rule	$\frac{a^m}{a^n} = a^{m-n}$	$\frac{x^5}{x^2} = x^3$
Power of a Power	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
Power of a Product	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
Power of a Quotient	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	$\left(\frac{x}{y}\right)^2 = \frac{x^2}{y^2}$

Additional info:

• These rules are foundational for simplifying algebraic expressions, solving equations, and working with polynomials and rational expressions.

• Mastery of negative exponents is essential for success in higher-level mathematics, including calculus and science courses.